# Full text of "Treatise On Analysis Vol-Ii"

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54 XII TOPOLOGY AND TOPOLOG1CAL ALGEBRA coset xH, and therefore {<*)} = n(xR) is open in G/H (12.10.6). Hence G/H is discrete, (iii) Let U0 be a neighborhood of e in G such that U0 n H = (e}> and let V0 be a symmetric open neighborhood of e such that Vg Q U0 (12.8.3). Then, for each x e G, the restriction of TC to V = xV0 is injective, because if h, ti in H are such that xzh = xz'ti with z, z' in V0» we have /z'/T1 = z/-1z 6 Vo c U0, so that h' = /z and z' = z. Since the image under TC of any open set in Y is open in n(V) (12.10.6), and n is continuous, it follows that the restriction of n to V is a homeomorphism onto 7i(V). (12.11.3) Let G be ametrizable group, H a closed subgroup ofG, and let dbe a right-invariant distance defining the topology of G (12.9.1). For any two points x, y in G/H, put d0(x, j>) = d(xR, jH) (3.4). Then d0 is a distance defining the topology o/G/H. IfG is complete (12.9.2), then G/H is complete with respect to dQ. We remark first that if x e x and yey, we have d0(x, y) = d(x>yH). For d(x, yJf) = inf d(x, yh) and therefore, for every h' e H, we have d(xh'9 j>H) = heH d(x9 yK) by virtue of the right-invariance of d. Hence, for each z e G, we have l</o(*, *) - <*o(y, 4)l = \*(x, *H) - d(y, zH)| g d(jc, y) by (3.4.2). Since this is true for all x, y in the respective cosets x, y, it follows that |rf0(x, z) - rf0(y» *)l ^ ^o(*> J?)* which shows that dQ is a pseudo-distance on G/H. The relation d0(x, y) < r is equivalent to the existence of a point y e y such that d(x, y) < r. Since H is closed in G, this proves that d0 is a dis- tance, and that if B(x; r) is the open ball with center x and radius r (with respect to d), then its image under the canonical mapping TC : G -> G/H is the open ball with center n(x) and radius r with respect to d0. Hence d0 defines the topology of G/H. Now suppose that G is complete. Let (xn) be a Cauchy sequence with respect to d0. We shall show that, by passing to a subsequence of (xw) if necessary, we can reduce to the case where there exists a right Cauchy sequence (xn) in G such that rc(xn) = xn. Since by hypothesis the sequence (xn) converges, this will show that (xn) has a cluster value and therefore converges (3.14.2), and hence that G/H is complete. There exists in G a denumerable fundamental system (Vn) of neighbor- hoods of e such that V^+1 <= Vn for all n (12.9.1 and 12.8.3). Let (BB) be a sequence of real numbers >0 such that the relation d(e, z) < sn implies that z e Vn. The hypothesis implies that we can define inductively a subsequence (xnj) of (xn) such that d0(xnp, xnq) < ek for all p ^ k and q ^ k. Replacing the original sequence (xn) by (xnjc), we can therefore assume that d0(xp ,xa) < enforclosed orbit Z (i.e., such that M • z = Z for all z e Z). (For each y e M • x